Does Workshop mana produced under Damping Sphere still have the "artifacts only" restriction?

@dr-j Yes

The amount of mana is changed but all riders on it remain the same.

This is also the case with other cards which have a similar effect such as Contamination.

Incidentally, Untaidake, the Cloud keeper can only be used to cast legendary spells even while there is a damping
sphere in play.

Alternatively, if you control an Urborg, Tomb of Yawgmoth or a Chromatic Lantern then both Mishra's Workshop and Untaidake, the Cloud Keeper may be used to cast non-artifact and non-legendary spells. This is because cards like Urborg, Tomb of Yawgmoth et al. grant the land an additional ability which does not have the same inherent rider associated with it.

Another interesting corner case is when a player has a Mana Flare or Wild Growth type of effect. Since the land that is tapped for mana is not the source of the additional mana it will not be affected by Damping Sphere; however, in the case of a card like Mana Reflection, which directly modifies the amount of mana a land produces, damping sphere will cause all of those lands to produce ◇ insteaad of any other type and amount.

Something worth noting is that if there are two replacement effects modifying the same mana ability it is up to the controller of that mana ability to decide what order the replacement affects are applied. In some cases this allows the controller of the mana ability to chose which replacement affect applies last and thus control the resultant outcome of the two replacement effects; however, there are also many occasions where that choice is not relevant to the outcome and the result will be the same no matter what order they are applied in. For example, if a player controls a Mishra's Workshop and there is a Damping Sphere and a Contamination on the battlefield then the controller of the Mishra's Workshop can chose the order in which to apply each replacement effect. If they chose to order Damping Sphere's effect first followed by Contamination's then Mishra's Workshop will be modified to add ◇ instead of 3 and then Contamination's effect will apply causing it to add instead of ◇. If they chose to order Contamination's effect first followed by Damping Sphere's then Mishra's Workshop will be modified to add instead of 3 after which Damping Sphere's effect will no longer apply since it isn't adding more than one mana anymore.

I'm not sure why you'd ever cast a sorceress, but can you walk me through the reason Boseiju can only cast them and instants when Damping is in play? I don't see why Damping would have any effect on Boseiju at all, since Boseiju doesn't tap for more than one mana. ?

@aaron-patten Thanks!

@topical_island Good Point. That was a poor example. There aren't that many other situation where there might be riders
on a land but, you're right, Boseiju doesn't get hit by sphere because it doesn't produce multiple mana. I've edited the original post.
Thanks for pointing it out.

Just another thing: Boseiju can normally cast any spell, but they won't be uncounterable if they aren't instants or sorceries.

@aaron-patten np. About 7/10 it's my mistake anyway. I was honestly asking, because for the life of me I couldn't figure out what I was missing there. Thanks for answering the original question though. I've been wondering the same thing, especially in relation to Damping and Energy Flux getting run against Shops at the same time.